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Secret
Secret
00:02
(cont. + corrections)
Tier 1:
$0 = \varnothing \in A$
$\forall n \in A : n^+=\{n,\{n,n\}\} \in A$
;
0 : Nat
Succ : Nat $\to$ Nat
True, False, Null : Bool
Truthvalue : Nat $\to$ Bool
Succ(Succ(0)) $\to$ Null
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2 hours later…
Secret
Secret
02:14
f: Nat $\times$ Nat $\to$ Bool
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Actually, start over again
ZF and stronger/weaker theories:
Tier 0:
$0 = \varnothing$
Tier 1:
$0 = \varnothing \in A$
$\forall n \in A : n^+ := \{n,\{n,n\}\} \in A$
$0^{++} \not\in A$
Result: $\{0,1\}$
Tier 2:
$0 = \varnothing \in A$
$\forall n \in A : n^+ := n \cup \{n\} \in A$
$\forall P[[P(0) \land \forall (n^+ \in A) [P(n^+)\implies P(n)]]\implies P(A)]$
$\forall (n\in A)[ \exists \omega > n]$
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$\forall P[[P(0)\land \forall (k \in A)[P(k^+) \implies P(k)]]\implies \forall (n \in A) P(n)]$
Let P(x)="$x \in A$"
$\forall n \in A \exists \omega [\omega > n \land \omega \not\in A]$
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Result: $\{0,1,2,3,4,5,6,...\}$
Tier 3:
$0 = \varnothing \in A$
$\forall n \in A: n^+ := n \cup \{n\} \in A$
$\forall P[[P(0)\land \forall (k \in A)[P(k^+) \implies P(k)]]\implies \forall (n \in A) P(n)]$
$\exists \omega = \{[0 \in A \land \forall (k \in A)[k^+ \in A \implies k \in A]]\implies n\in A\}$
$\omega \in C$
$B = A \cup \{[\omega \in C \land \forall (\beta \in C)[\beta^+ \in C \implies \beta \in C]]\implies \alpha\in C\}$
Result: $\{0,1,2,3,4,5,6,...,\omega,\omega+1,\omega+2,\omega+3,\omega+4,\omega+5,\omega+‌​6\}$
Tier 4:
 
10 hours later…
Secret
Secret
13:22
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